Question

How can you memorize exponent rules?

Answer 1

See explanation...

Explanation:

Start with positive integer exponents:

`a^n = overbrace(a xx a xx ... xx a)^"n times"`

Then you can see:

`a^m xx a^n = overbrace(a xx a xx ... xx a)^"m times" xx overbrace(a xx a xx ... xx a)^"n times"`

`=overbrace(a xx a xx ... xx a)^"m + n times" = a^(m+n)`

This is useful when you multiply two numbers that are expressed in scientific notation. For example:

`(1.2 xx 10^3) xx (2.4 xx 10^6)`

`=(1.2 xx 2.4) xx (10^3 xx 10^6)`

`=2.88 xx 10^(3+6)`

`=2.88 xx 10^9`

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For negative exponents, first note that if `a != 0`:

`a^(-n) = 1/underbrace(a xx a xx ... xx a)_"n times"`

and we find:

`a^n xx a^(-n) = overbrace(a xx a xx ... xx a)^"n times" xx 1/underbrace(a xx a xx ... xx a)_"n times"`

`=overbrace(a xx a xx ... xx a)^"n times"/underbrace(a xx a xx ... xx a)_"n times" = 1`

We find that the rule: `a^m xx a^n = a^(m+n)` works for any integer values of `m` and `n`, positive, negative or `0`.

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The next level of complexity is:

`(a^m)^n = overbrace(a^m xx a^m xx .. xx a^m)^"n times" = a^(mn)`

For example:

`(2^2)^3 = 4^3 = 64`

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Finally note that `a^(m^n)` is evaluated from right to left.

That is:

`a^(m^n) = a^((m^n))`

For example:

`2^(2^3) = 2^8 = 256`